By Christian Peskine
Peskine does not supply loads of causes (he manages to hide on 30 pages what frequently takes up part a ebook) and the routines are difficult, however the e-book is however good written, which makes it beautiful effortless to learn and comprehend. urged for everybody keen to paintings their approach via his one-line proofs ("Obvious.")!
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Extra info for An Algebraic Introduction to Complex Projective Geometry: Commutative Algebra
We have S-'K = S-'C = (0). Since A is Noetherian, the submodule K of nA is finitely generated. Hence there exist t and U in S such that Kt = (0) and C, = (0). If we put s = tu, we have K, = C, = (0), hence f, is an isomorU phism and M , is a free A,-module of rank n. 24 (i) If M and N are A-modules and S a multiplicatively closed part of A , there is a natural homomorphism of S-'A-modules: S-'HOmA(M, N ) + HOmS-1A(S-'M1 S - l N ) . (ii) If III is finitely generated, this homomorphism is injective.
We can now prove, by induction on ~ A ( M )that , the evaluation homomorphism eD,M : M HomA (HOmA ( M ,D ), D ) This induces an exact sequence 0 ---f HomA(M, D ) + HomA(n-4, D ) 4 HomA(K, D ) -+ is an isomorphism for all finitely generated modules M . Assume l ~ ( h l > ) 1. Let M' c M be a strict submodule. We have l ~ ( h f ' )< ~ A ( Mand ) ~A(M/M')< ~ A ( M )Consider . the following commutative diagram (where we write Ni" for HomA (HomA( N ,D ), D ) ) : which shows lA(HomA(nA, D ) ) 5 1A(HOmA(M,D ) ) 4-~ A ( H O ~ A D ( K) ), Since HomA(nA, D ) = nD, we have lA(HomA(nA,D ) ) = nlA(D) = n l ~ ( A= ) 1 ~ ( h lf) ~ A ( K ) , ~ A ( Mf) ~ A ( K5) 1A(HOmA(M,D ) ) f ~ A ( H ~ ~ D A )() K 5 l, A ( M )f lA(K), 1A(HomA(M,D ) ) = lA(M).
3 (4), we can enlarge our commutative diagram as fol- 2. The length, defined in the category of finite length A-modules, with value in Z,is a n additive function. 7 Let A be a principal ideal ring. If M is a finitely generated A-module, we recall that M / T ( M ) (where T ( M ) is the torsion submodule of M ) is a free A-module. We define rkA(M) = rk(M/T(M)). Show that rkA(*) is a n additive function on the category of finitely generated A-modules. The following property of additive functions is practically contained in their definition.
An Algebraic Introduction to Complex Projective Geometry: Commutative Algebra by Christian Peskine